Example:
A rectangle is 4 times as long as it is wide. If the length is increased by 4 inches and the width is decreased by 1 inch, the area will be 60 square inches. What were the dimensions of the original rectangle?
Solution:
Step 1: Assign variables:
Let x = original width of rectangleStep 2: Write out the formula for area of rectangle.
A = lwStep 3: Plug in the values from the question and from the sketch.
60 = (4x + 4)(x –1)
Use distributive property to remove brackets
60 = 4x2 – 4x + 4x – 4
Put in Quadratic Form
4x2 – 4 – 60 = 0
4x2 – 64 = 0
This quadratic can be rewritten as a difference of two squares
(2x)2 – (8)2 = 0
Factorize difference of two squares
(2x)2 – (8)2 = 0
(2x – 8)(2x + 8) = 0
Since x is a dimension, it would be positive. So, we take x = 4
The question requires the dimensions of the original rectangle.
The width of the original rectangle is 4.
The length is 4 times the width = 4 × 4 = 16
Answer: The dimensions of the original rectangle are 4 and 16.
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